Question

Let $f$ be a function defined as $f\left(x\right)=\frac{3+2\sin x}{9-4{\sin }^{2}x}.$ Then which of the following option(s) is (are) CORRECT?

• A. Domain of $f$ is $R$
• B. Domain of $f$ is $R-\left\{\frac{3}{2}\right\}$
• C. Range of $f$ is $[\frac{1}{5},1]$
• D. Range of $f$ is $[\frac{1}{3},1]$

Answer 1

Answer: (A), (C)

The correct options are
A Domain of $f$ is $R$
C Range of $f$ is $[\frac{1}{5},1]$

$f\left(x\right)=\frac{3+2\sin x}{9-4{\sin }^{2}x}$
$=\frac{3+2\sin x}{(3+2\sin x)(3-2\sin x)}$
$=\frac{1}{3-2\sin x}(\because 3+2\sin x\ne 0,\forall x\in R)$
$\Rightarrow f\left(x\right)=\frac{1}{3-2\sin x}$
Denominator $>0$ for all real values of $x$
Since, $f\left(x\right)$ is defined for all $x\in R$
$\therefore$ Domain is $R$

We know that
$-1\le \sin x\le 1$
$\Rightarrow -2\le -2\sin x\le 2$
$\Rightarrow 1\le 3-2\sin x\le 5$
$\Rightarrow \frac{1}{5}\le \frac{1}{3-2\sin x}\le 1$
$\therefore$ Range is $[\frac{1}{5},1]$